We will discuss Section 3 (pages 25-29) proving associativity of elliptic curve addition.

As time permits, we will discuss the statement of the Riemann Roch theorem.

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# Pitt Math 2810 – Algebraic Geometry – Spring 2016

## A graduate course in Algebraic Geometry

# Friday April 27

# Wednesday April 27

# Monday April 25

# Friday April 22

# Wednesday April 20, 2016

# Monday April 18, 2016

# Friday, April 15, 2016

We will discuss Section 3 (pages 25-29) proving associativity of elliptic curve addition.

As time permits, we will discuss the statement of the Riemann Roch theorem.

- Read up through page 11 completing the section on intersection numbers. The intersection number is the dimension of a vector space. Why is that vector space finite dimensional?
- Review projective plane curves, translating what Milne does back into the language of Proj(S.).
- Learn the statement of Bezout’s theorem.
- Read curves of degree two. Is it really true that C can’t be singular?
- Read up to the end of page 20.

The final week of meetings will go over introductory material on elliptic curves from Milne’s book

Read the introduction about some of the uses of elliptic curves (including sphere packings and cryptography!)

pages 5-6 should be review.

- Read the definition of a nonsingular curve.
- Work example 1.5 to get a test for a Weierstrass polynomial to define a nonsingular curve.
- Read Prop 1.8 on intersection numbers.
- Read up through page 11 completing the section on intersection numbers
- Review projective plane curves, translating what Milne does back into the language of Proj(S.).
- Learn the statement of Bezout’s theorem.

- We will discuss a later result (8.6) from Humphreys: let G be an affine algebraic group over an algebraically closed field k, then G is isomorphic to a closed subgroup of GL(n) (over k) for some n.
- 7.3. Let G be an affine algebraic group. Show that G^0 is a normal subgroup of finite index.
- 7.4. Show that the Zariski closure of a subgroup is a subgroup.
- 7.4.B. Show that the kernel and images are closed subgroups, and that the image of the connected component of the identity is the connected component of the image.
- 8.1. State what an action is in the category of k-varieties.
- Show that GLn acts on P^n, on the flag variety.

We will start on our discussion of elliptic curves as time permits.

- Read the section 6 on complete varieties (pages 45-46) and prepare to discuss Proposition 6.1.
- Read and prepare the proof that P^n is complete.
- Learn the definition of G_m, G_a, GL(n), SL(n), O(n), SO(n), Sp(2m) as algebraic groups. In each case, describe the morphism from A to A\otimes_k A that is dual to the multiplication mu:GxG->G, where G = Spec(A).
- 7.3. Let G be an affine algebraic group. Show that G^0 is a normal subgroup of finite index.
- 7.4. Show that the Zariski closure of a subgroup is a subgroup.
- 7.4.B. Show that the kernel and images are closed subgroups, and that the image of the connected component of the identity is the connected component of the image.
- 8.1. State what an action is in the category of k-varieties.

- Read 6.4. (automorphisms of the projective line) and prepare the proof that that the automorphism group is PGL(2,k).
- Read the section 6 on complete varieties (pages 45-46) and prepare to discuss Proposition 6.1.
- Read the proof that P^n is complete.
- Show how the product of two elliptic curves has the form Proj(S_.).
- Learn the definition of algebraic group and abelian variety.

The meeting will continue with Humphrey’s book.

- Work through the details of the construction of the Grassmannian as a k-scheme.
- Work the the construction of the flag variety
- Read 6.4. (automorphisms of the projective line) and prepare the proof that that the automorphism group is PGL(2,k).
- Read the section 6 on complete varieties (pages 45-46).